可壓縮顆粒流μ(I)與Φ(I)基底流變模型之理論與數值研究

Abstract

顆粒流展現出多相的流動模式，包括以碰撞為主的類氣體區、以摩擦為主的類固體區，以及碰撞與摩擦同時作用的類液體區。本論文旨在分析類液體區的流變模型穩定性，並對不同流場進行連續體模擬。其中，討論了經典的μ(I)和Φ(I)流變模型在完全發展流中的穩定性，並以無因次參數χ作為穩定性的判斷標準；此外，也引入近年提出的CIDR和iCIDR流變模型，推導降伏函數和膨脹函數的解析解，並以顯式形式表示正向壓力和偏剪應力。

本研究也在卡氏坐標和極坐標系中開發了iCIDR 模型的數值模擬方法，通過無因次化統御方程式，將其表示為保守形式，形成一個混合雙曲—拋物系統，並應用分裂法將其分為雙曲系統和拋物系統兩步驟。透過模擬完全發展剪切流、斜坡流，以及軸對稱環形剪切流的暫態流況，與解析解和文獻的實驗與模擬結果相互驗證，確認方法論的可用性。最後，運用超限插值法定義曲線坐標系，將複雜的物理區域轉化為精簡的計算區域，保持數值近似的準確性，並以模擬具有弧形上板的二維剪切流的暫態發展過程來進行收斂性驗證。

Granular flow exhibits multiphase patterns: gas-like (collision-dominated), solid-like (friction-dominated), and liquid-like (both collision and friction). This thesis analyzes the stability of the rheological model in the liquid-like regime and performs continuum simulations of various flow fields. We examine the stability of classical μ(I) and Φ(I) rheology models in fully-developed flow using the dimensionless parameter χ. The CIDR and iCIDR rheology models are introduced with analytical solutions for yielding and dilation functions, explicitly expressing normal pressure and deviatoric shear stress.

Numerical methods for the iCIDR model in Cartesian and polar coordinates are developed, representing the dimensionless governing equations in a conservative form to create a mixed hyperbolic-parabolic system, solved by the fractional step method. Validity is verified by simulating transient responses of fully-developed simple shear, fully-developed inclined surface, and axisymmetric annular shear flows. Furthermore, the transfinite interpolation method is applied to simplify complex physical domains while maintaining numerical accuracy. The transient development of a shear flow with an arched upper plate is simulated for convergence verification.